Two metal wires of identical dimensions are connected in series. If σ$$1$$ and σ$$2$$ are the conductivities of the metal wires respectively, the effective conductivity of the combination is

Question

Two metal wires of identical dimensions are connected in series. If σ$$1$$ and σ$$2$$  are the conductivities of the metal wires respectively, the effective conductivity of the combination is

Infinity Learn · Accepted Answer

As both metal wires are of identical dimensions, so their length and area of $$cross-section$$ will be same. Let them be I and A respectively. Then The resistance of the first wire $$isR1=l$$σ$$1A$$                                ....(i)and$$ that of the second wire is $$R2=l$$σ$$2A$$                       ...$$(ii)As$$ they are connected in series, so their effective resistance $$isRs=R1+R2=l$$σ$$1A+l$$σ$$2A$$  $$($$ using $$(i) and  (ii))=lA1$$σ$$1+1$$σ$$2$$                  . . .$$(iii)If$$ σeff is the effective conductivity of the combination, $$thenRs=2l$$σeff A                        . . . .$$(iv)Equating$$ eqns. $$(iii) and (iv), we get $$2l$$σ$$effA=lA1$$σ$$1+1$$σ$$22$$σeff $$=$$σ$$2+$$σ$$1$$σ$$1$$σ$$2$$⇒σeff $$=2$$σ$$1$$σ$$2$$σ$$1+$$σ$$2$$

Two metal wires of identical dimensions are connected in series. If σ1 and σ2 are the conductivities of the metal wires respectively, the effective conductivity of the combination is

Detailed Solution

Get Expert Academic Guidance – Connect with a Counselor Today!